diff --git a/EDIT2013_Kondo_ans/Makefile b/EDIT2013_Kondo_ans/Makefile new file mode 100644 index 0000000..5dfad37 --- /dev/null +++ b/EDIT2013_Kondo_ans/Makefile @@ -0,0 +1,5 @@ +all: + pdflatex kondo_ex.tex + +clean: + rm -rf *.aux kondo_ex.pdf *.log diff --git a/EDIT2013_Kondo_ans/kondo_ex.tex b/EDIT2013_Kondo_ans/kondo_ex.tex new file mode 100644 index 0000000..0ec2acd --- /dev/null +++ b/EDIT2013_Kondo_ans/kondo_ex.tex @@ -0,0 +1,168 @@ +\documentclass[a4paper]{article} +\usepackage[utf8x]{inputenc} +\usepackage{ucs} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{epsf} +\usepackage{float} +\usepackage{url} +\usepackage{fancyhdr} +\usepackage{verbatim} +\usepackage{color,listings} +\usepackage{booktabs} +\usepackage{tabularx} +\usepackage{natbib} +\usepackage[all]{xy} +\usepackage{graphicx} +\usepackage{graphics} +\usepackage{color} +\usepackage{listings} + +%\makeatletter +%\def\@xobeysp{} +%\makeatother +\usepackage{color,listings} + +\author{Tran Hoai Nam} +\title{Answer to quizzes in \\Detectors of LHC Experiments lecture} + +\begin{document} +\maketitle +\section*{Quiz 1: Number of protons for $\mathcal{L}=10^{34}$ +cm$^{-2}$s$^{-1}$} +Total luminosity of LHC: +\[ +\mathcal{L} = f N_b L_{bunch\_cross} = f N_b \frac{N_p \times N_p}{4\pi\sigma_x\sigma_y} +\] +\[ +\rightarrow N_p = \sqrt{ 4\pi \sigma_x\sigma_y \frac{\mathcal{L}}{fN_b}} +\] +where: $f$ is revolution frequency; +\[ +f \approx \frac{c}{l_{LHC}} += \frac{3\times 10^8 [ms^{-1}]}{27 \times 10^3 [m]} \approx 10^4 [s^{-1}] +\] +Substitute the numbers, number of proton in each bunch is: +\[ +N_p = +\sqrt{4\times 3.14 \times 16 \times 10^{-6} [m]\times 16 \times 10^{-6} [m] +\times +\frac{10^{34} \times 10^{4}[m^{-2}s^{-1}]}{2808 \times 10^4[s^{-1}]}} +\approx 1.07 \times 10^{-11} +\] + +\section*{Quiz 2: Characteristics of a silicon detector} +\subsection*{1) Full depletion voltage} +Let $V$ be the full depletion voltage for the 300 $\mu m$ Si detector: +\[ +d = \sqrt{\frac{2\varepsilon^{'} \varepsilon_0}{q_e N_{eff}}V} +\rightarrow V = \frac{q_e N_{eff}}{2 \varepsilon^{'} \varepsilon_0} d^2 +\] +\[ +V = \frac{1.6 \times 10^{-19} [C] \times 2 \times 10^{12} [cm^{-3}]} +{2\times11.6\times 8.854 \times 10^{-14} [F cm^{-1}]} (300 \times 10^{-4} [cm])^2 +\approx 140.2 [V] +\] +\subsection*{2) Noise} +Capacitance of the detector when it is fully depleted: +\begin{align*} +C &= \varepsilon^{'} \varepsilon_0 \frac{S}{d} \\ + &= 11.6 \times 8.854 \times 10^{-14} [F cm^{-1}] + \frac{1 [cm^2]}{300 \times 10^{-4} [cm]} \\ + &\approx 34.2 \times 10^{-12} [F] \\ + &= 34.2 [pF] +\end{align*} + +Equivalent noise charge square: +\[ +ENC^2 = \left(\frac{e^3}{36q_e}\right)^2 +\left[ \frac{3}{\tau} (4k_B T R_S) C^2 + \frac{5\tau}{3}(2q_eI_{leak})\right] +\] + +We have: +\begin{align*} +k_B T &= 8.62 \times 10^{-5} [eV K^{-1}] \times 273.15 [K] \\ + &\approx 2.35 \times 10^{-2} [eV] \\ + &= 2.35 \times 10^{-2} \times 1.6 \times 10^{-19} [J]\\ + &= 2.35 \times 10^{-2} q_e [JC^{-1}] +\end{align*} +Thus: +\begin{align*} +ENC^2 = + & \frac{2.72^6}{36^2 \times 1.6 \times 10^{-19}[C]} \times \\ + & \times \Biggl[ + \frac{3}{20 \times 10^{-9} [s]} + (4 \times 2.35 \times 10^{-2} [JC^{-1}] \times 200 [\Omega]) + (34.2 \times 10^{-12} [F])^{2} + \\ + & + \frac{5 \times 20 \times 10^{-9}[s]}{3} + (2 \times + 10 \times 10^{-6} [A]) + \Biggr] +\end{align*} + +\begin{align*} +ENC^2 &= + \frac{2.72^6}{36^2 \times 1.6 \times 10^{-19}[C]} + \times \Biggl[ + 3.3 \times 10^{-12}[Js^{-1}C^{-1}\Omega F^2] + 0.3 \times 10^{-12} [C] + \Biggr] \\ + &\approx 7.0\times 10^6 +\end{align*} + +(dimensionally speaking, $Js^{-1}C^{-1}\Omega F^2 += VA \times C^{-1}\times VA^{-1}\times C^2V^{-2} = C$) + +\[ +\rightarrow ENC \approx 2.65 \times 10^3 +\] + +The $ENC$ is $2.65 \times 10^3$, and it can be seen that the contribution from +capacitance is dominant. +%\begin{align*} +%ENC^2 = + %&\left( + %\frac{2.72^3}{36\times 1.6\times 10^{-19}[C]} + %\right)^2 \times \\ + %& \times \Biggl[ + %\frac{3}{20 \times 10^{-9} [s]} + %(4 \times 8.62 \times 10^{-5} [eV K^{-1}] + %\times 273.15 [K] \times 200 [\Omega]) + %(34.2 \times 10^{-12} [F])^{2} + \\ + %& + \frac{5 \times 20 \times 10^{-9}[s]}{3} + %(2 \times 1.6 \times 10^{-19}[C] \times + %10 \times 10^{-6} [A]) + %\Biggr] +%\end{align*} + +%\begin{align*} +%ENC^2 = + %&\left( + %\frac{2.72^3}{36\times 1.6\times 10^{-19}[C]} + %\right)^2 \times \\ + %& \times \Biggl[ + %\frac{3}{20 \times 10^{-9} [s]} + %(4 \times 2.35 \times 10^{-2}[eV] \times 200 [\Omega]) + %(34.2 \times 10^{-12} [F])^{2} + \\ + %& + \frac{5 \times 20 \times 10^{-9}[s]}{3} + %(2 \times 1.6 \times 10^{-19} [C]\times + %10 \times 10^{-6} [A]) + %\Biggr] +%\end{align*} + +%\begin{align*} +%ENC^2 = + %&\left( + %\frac{2.72^3}{36\times 1.6\times 10^{-19}[C]} + %\right)^2 \times \\ + %& \times \Biggl[ + %\frac{3}{20 \times 10^{-9} [s]} + %(4 \times 2.35 \times 10^{-2} \times 1.6 \times 10^{-19}[J] \times 200 [\Omega]) + %(34.2 \times 10^{-12} [F])^{2} + \\ + %& + \frac{5 \times 20 \times 10^{-9}[s]}{3} + %(2 \times 1.6 \times 10^{-19} [C]\times + %10 \times 10^{-6} [A]) + %\Biggr] +%\end{align*} + +\end{document}